Question

Determine a point which divides a line segment of length 12 cm internally in the ratio 2 : 3 Also, justify your construction.

Answer 1

Given that

Determine a point which divides a line segment of length 12cminternally in the ratio of 2:3.

We follow the following steps to construct the given

Step of construction

Step: I- First of all we draw a line segment AB = 12 cm.

Step: II- We draw a ray AX making an acute angle ∠BAX = 60° with AB.

Step: III- Draw a ray BY parallel to AX by making an acute angle ∠ABY = ∠BAX.

Step IV- Mark of two points A₁, A₂ on AX and three points B₁, B₂, B₃ on BY in such a way that AA₁ = A₁A₂ = BB₁ = B₁B₂ = B₂B₃.

Step: V- Joins A₂B₃ and this line intersects AB at a point P.

Thus, P is the point dividing AB internally in the ratio of 2:3

Justification:

In ΔAA₂P and ΔBB₃P we have

∠A₂AP = ∠PBB₃ [∠ABY = ∠BAX]

And ∠APA₂ = ∠BPB₃    [Vertically opposite angle]

So, AA similarity criterion, we have

ΔAA₂P ≈ ΔBB₃P

`(A A_2)/(BB_3)=(AP)/(BP)`

`(AP)/(BP)=2/3`